GIRD Systems CORDIC Tutorial

http://www.girdsystems.com

Using the CORDIC Algorithm

for Fast Trigonometric

Operations in Hardware

GIRD Systems Internal Seminar

November 25, 2009


What is the CORDIC algorithm?

• COordinate Rotation DIgital Computer

• Fast way to compute:

– Magnitude and phase of a complex value

– Trigonometric functions such as sin and cos

• Originally developed in 1959 for real-time aircraft navigation computer

• Computes values based on a series of rotations


Why use the CORDIC algorithm?

• Major reason – only multiplications required are bit shifts (simple in hardware)

• Multiplier units are resource hogs

• Iterative process – more iterations yields better accuracy

– Easy to trade off execution speed vs. accuracy

– Design to the constraints of the system


When to use CORDIC?

• Any application that requires rapid calculation of trigonometric functions – Demodulation of successive carriers in OFDM symbols

– Rectangular-to-polar conversion operations

• Especially useful for FPGA implementations with space constraints – CORDIC blocks are available from many

manufacturers such as Xilinx


CORDIC Algorithm • Apply a series of

rotations in the positive or negative direction

• Rotation is in increments of arctan(2-L) : L = 0 45° L = 1 26.565° L = 2 14.036° L = 3 7.125°

• Values can be precomputed and stored


• Mathematical view of the approach:

Start with a complex value: C = I

c+jQ

c

And a rotational value: R = Ir+jQ

r

To add a rotation to C: C' = C•R To subtract a rotation: C' = C•R*

CORDIC Algorithm


Adding a Rotation • For the CORDIC algorithm R = 1 + jK where K = 2-L

• Breaking out the equation for adding an equation:

C' = C•R Ic' = I

cIr – Q

cQ

r = I

c – (Q

c >>L)

Qc' = Q

cIr + I

cQ

r = Q

c+ (I

c>>L)

• Special case: Adding 90° R = 0 + j1 I

c' = -Q

c

Qc' = I

c (negate Q

c and swap)


Subtracting a Rotation • For the CORDIC algorithm R* = 1 - jK where K = 2-L

• Breaking out the equation for adding an equation:

C' = C•R* Ic' = I

cIr + Q

cQ

r = I

c + (Q

c >>L)

Qc' = Q

cIr - I

cQ

r = Q

c - (I

c>>L)

• Special case: Subtracting 90° R = 0 - j1 I

c' = Q

c

Qc' = -I

c (negate I

c and swap)


Computing sin and cos

• Step 1: Start at Ic = 1 and Q

c = 0

• Step 2: Rotate in increments of 90° to move into the correct quadrant

• Step 3: Iterate over L, starting at L=0, until the desired accuracy is achieved. Add or subtract rotations so that the sum converges on the desired angle

• Ic = cos(b) Q

c = sin(b)


Important Caveat • Since the magnitude of the rotation vector is not unity

magnitude, we need to divide our results by a term called the CORDIC gain.

• CORDIC gain = prod(abs(1+j/2L),0,L)

• This term can be precomputed and stored

L = 0 Cg = 1.4142 1/C

g = 0.7071

L = 1 Cg = 1.5811 1/C

g = 0.6325

L = 2 Cg = 1.6298 1/C

g = 0.6136

• Cg converges to approximately 1.647


Computing sin and cos – A Simple Example

• Find the sin and cos of 108.435°

• Start at 0° C = 1 + j0

• Add 90° C = 0 + j1

• Add 45° (L=0,b=135°) C = (0-1) + j(1+0) = -1+j

• Subtract 26.565° (L=1,b=108.435°)

C = (-1 + 1/2) + j(1-(-1)/2) = -0.5 + j1.5

• Divide by the CORDIC gain for L = 1, 1/Cg

= 0.6325

• sin(108.435°) = 1.5(0.6325) = 0.9487 cos(108.435°) = -0.5(0.6325) = -0.3162


Computing sin and cos – Another Example

• Find the sin and cos of 292.2°

• Start at 270° C = 0-j

• Add 45° (L=0, b=315°) C = 1-j

• Subtract 26.565° (L=1,b=288.435°) C = 0.5 - j1.5

• Add 14.036° (L=2, b=302.471°) C = 0.875 - j1.375

• Subtract 7.125° (L=3,b=295.346°) C = 0.703 - j1.484

• Subtract 3.576° (L=4,b=291.767°) C = 0.610 - j1.528

• Add 1.790° (L=5,b=293.560°) C = 0.658 – j1.509

• Subtract 0.895° (L=6,b=292.665°) C = 0.635 - j1.520

• Add 0.448° (L=7,b=292.217°) C = 0.623 - j1.524


Computing sin and cos (cont.) • We are pretty close to 292.2°, so we will stop here

• C = 0.623 - j1.524

• Divide by CORDIC gain Ic = cos(292.217°) = 0.623/1.64674 = 0.3781

Qc= sin(292.217°) = -1.524/1.64674 = -0.9258

• Compare to the actual values: cos(292.2°) = 0.3778 sin(292.2 °) = -0.9259

• Result will improve as more iterations are used


Matlab Code for Previous Example beta = 270; phi = 292.2; C = 0-j; %initialize terms

num_iter = 7; angles = 180/pi.*atan(2.0.^[0:-1:-1*num_iter]); %precompute CORDIC angles

for N = 1:length(angles)

L = N-1 % print out L

if (beta < phi) % add a rotation if beta < phi

beta = beta + angles(N)

C = C*(1+(j/(2^(N-1))))

elseif (beta > phi) % subtract rotation if beta > phi

beta = beta - angles(N)

C = C*(1-(j/(2^(N-1))))

end

end

Cg = prod(abs(1+j./2.^[0:length(angles)-1])) %compute the CORDIC gain

C = C/Cg %cos(beta) = real portion, sin(beta) = imaginary portion


Computing Magnitude and Phase • To compute magnitude and phase, we work backwards

– start with a complex value and rotate until the Q component is 0 and the I component is positive (i.e. 0° angle)

• Like sin and cos, magnitude and phase are computed simultaneously

• Again, we have to factor in the CORDIC gain when computing the magnitude


Computing Magnitude and Phase – A Simple Example

• Use CORDIC to estimate the mag. and phase of -1+j3

• Since this number is in Quadrant II, we need to subtract 90°to get back to Quadrant I: C = 3+j1

• Now since I and Q are positive, we subtract the 45°term: C = (3+1>>0) + j(1-3>>0) = 4-j2

• Q is now negative, so we add the next phase term (26.57°): C = (4-(-2)>>1) + j(-2+4>>1) = 5+j0

• Now we are located on the real axis. The phase is the inverse of the sum of the rotations we applied: -1*(-90-45+26.57) = 108.43°


Computing Magnitude and Phase – A Simple Example (cont.)

• To compute the magnitude, we take the real result and divide by the CORDIC gain

• Real result = 5

• CORDIC gain for L = 1 is 1.5811

• 5/1.5811 = 3.162

• Compare to the actual result

abs(1+j*3) = sqrt(10) = 3.162


Computing Magnitude and Phase – Another Example

• Find the magnitude and phase for -3.84-j2.51

• Add 180°to move to Quadrant I : C = 3.84+j2.51

• Subtract 45°: C = 3.84+(2.51>>0) + j(2.51-3.84>>0) C = 6.35 - j1.33

• Add 26.565°: C=6.35-1.33>>1+j(1.33+6.35>>1) C=7.02+j1.85

• Subtract 14.036°: C=7.02+1.85>>2+j(1.85-7.02>>2) C=7.48+j0.09

• Q term is almost zero, so we will use this approximation

• Phase = -1*(180-45+26.565-14.036) = -147.529°

• Magnitude = 7.476/1.630 = 4.5872

• Compare to actual answer : 4.5876 @ -146.83°


Summary • CORDIC is a fast way to compute trig functions and

phase/magnitude in hardware

• Almost no multiplication or division required other than bit shifts

• Iterative process – more iterations helps zero in on a more correct result

• CORDIC functions are available for FPGAs and the algorithm can be incorporated into other devices such as fixed point DSPs

• Good for resource constrained platforms, but computing values directly is more accurate if hardware is able to meet speed requirements

Documents

GIRD Systems CORDIC Tutorial